090126HomeWork3

# 090126HomeWork3 - Y z = A z X-z i.e only the alias term...

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1 ECE 700 Digital Signal Processing Home Work 3 Due 2/4/09 1. Uniform DFT filterbanks (Vaidyanathan) Consider a uniform DFT filterbank with M = 5. Assume that the analysis filters are given by 2 /5 0 5 5 ( ) ( ), 0,1, ,4 where . k k j k H z H zW k W e π - = = = Let 1 1 4 2 2 3 ( ) ( ) ( ) and ( ) ( ) ( ). G z H z H z G z H z H z = + = + Assume that h 0 [ n ] is real. a. Are [ ],1 4 k h n k real for all n ? b. Express 1 2 [ ] and [ ] g n g n in terms of 0 [ ]. h n c. Are 1 2 [ ] and [ ] g n g n real for all n ? d. Let 0 ( ) j H e ϖ be as shown below. Sketch 1 2 ( ) and ( ) . j j G e G e 2. (Haar basis.) Consider the two channel QMF filterbank. Suppose 1 0 1 ( ) (1 ). 2 H z z - = + What choices for 1 0 1 ( ), ( ), and ( ) H z G z G z will result in a perfect reconstruction QMF filterbank? 3. (Filterbanks) Consider a two channel filterbank. We saw that we can write Y ( z ) = T ( z ) X ( z ) + A ( z ) X (- z ). Is it possible to choose the 4 filterbank filters so that

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Unformatted text preview: Y ( z ) = A ( z ) X (-z ) i.e. only the alias term gets through? If yes, what condition needs to be satisfied for this to be true? 2 4. (Mitra, problem 13.40) Consider the system shown below. a) Show that the overall system is alias free, and calculate the transfer function ( ) ( ) ( ) Y z T z X z = in terms of H ( z ) and H 1 ( z ). b) What is T ( z ) if 2 2 1 1 1 ( ) [ ( ) ( )], ( ) [ ( ) ( )]. 2 2 H z H z H z H z z H z H z = +-=--c) If H ( z ) is a length 2 K filter, what are the lengths of the filters H ( z ) and H 1 ( z )? d) How many multiplies and adds are required to compute one output sample if the length of H ( z ) is 2 K samples?...
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090126HomeWork3 - Y z = A z X-z i.e only the alias term...

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